Acoustics of Music

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Acoustics of Music

• Dr Ian Drumm

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Phase Vocoder

• Aims
• To introduce the concept of analysis synthesis
  methods
• Learning Outcomes
• Short-Time Fourier transform (STFT)
• The application of the phase vocoder

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Introduction

• The phase vocoder is an analysis synthesis
  technique that can also be realised as a
  sophisticated effects box
• Transforms signal into time-varying spectra.
  These can be altered and a new signal synthesised.

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Basic Concept

• The Discrete Fourier Transform

is a time series
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The Discrete Inverse Fourier Transform
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Phase Vocoder

• Take a signal (i.e. a signal indexed by k)
• Apply short time, windowed DFTs (or FFTs) to
  successive sections of length M
• Hop every by s samples after each section
• Store DFTs an array indexed by integers m,l

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Short Time Fourier Transform

• This gives the Short Time Fourier Transform
• Where
• M is the window length
• s is the hop size
• l is an integer (0,1,2,.L)
• h is the impulse response of the window
• The STFT is for Xm,l for all l (and m)
• Note each element of the STFT has frequency

Were fs is the sample rate
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In Matlab
clear all close all x,FS,NBITSwavread('bassoon.
wav') M4092 Window length sM/8 Hop
size sectionslength(x)/s - M/s Xzeros(M,sectio
ns) xwindowedzeros(M,1) for l0sections
xwindowedx((ls1)(lsM)) . hamming(M)
specabs(fft(xwindowed)) X(,l1)spec end
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Choosing Hop Size

• If hop size equals window size then half the
  signal will be attenuated
• If the overlap point corresponds to -3dB then the
  subjective importance of any attenuation will be
  very small.
• Typically chose some fraction of window size,
  e.g. M/8
• Decimation Factor e.g. a decimation factor of 2
  will have two overlaps in the window.

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Re-synthesis

• Put the musical signal back together again from
  the STFTs

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Re-synthesis

• Inverse STFTs
• where is now the part of the new time
  signal valid for frame l
• And knit them all together with the overlap-add
  method
• Where is the final output signal

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Time expansion or compression without pitch
distortion

• Vary the value of s on playback

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Pitch shifting without time distortion

• Spectrum scaling - Multiply all fm by scale
  factor (or by a function)
• Pitch shifting important tool in recording studio
  for correcting poor vocal performance

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Tricks with a phase vocoder Vary of recorded
sound

• Can modify the amplitude and frequency content of
  the STFTs in pretty much anyway you like
• E.g. the phase vocoder can act as a very flexible
  and accurate filter by varying to amplitude of
  the values in the frequency bins of the STFTs.

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Other creative applications

• Timbral interpolation from one instrument to
  another by interpolating between STFTs
• Cross-synthesis using a characteristic from one
  sound (amp, freq or phase) to modulate a
  characteristic from another.

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Analogy of Subtractive Synthesiser Vocoders

• Included with popular synths (Moogs, Korgs, etc)
• Array of analog band pass filters that take say a
  voice signal and split it up into a range of time
  varying amplitudes corresponding to different
  frequencies (a crude STFT).
• These time varying amplitudes can then be used to
  control another signal (carrier) such as a
  synthesiser timbre or an electric guitar sound.
• ELOs Mr Blue Sky

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Parameters and choices

• Window size M larger means good freq resolution,
  poor time resolution, long calculation time
• Window type anything non-rectangular (Hamming,
  Hanning, Blackman, etc)
• FFT size Optimisation restricts to 2n e.g. 1024
• Hop size, s fraction of M e.g. sM/8 overlap
  at -3dB points of window

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Problems

• Transients not dealt with effectively
• The impulse response of window convolves signal
• Heisenberg uncertainty - We cannot exactly know
  what frequency exists at what time instance , we
  can only know what frequency bands exist at what
  time intervals
• Higher frequencies are better resolved in time
  i.e. a HF signal can be located in time better
  than frequency
• Lower frequencies are better resolved in
  frequency i.e. a LF signal is located in
  frequency better than time.

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Improvements

• Window closing
• Repeat analysis of signal xk with smaller and
  smaller window sizes
• Tracking phase vocoder
• Find peaks in magnitude spectrum and attempt to
  track these through successive spectra (producing
  an envelop for each peak) discard other
  magnitude information
• Different Transforms
• Reduce effects of time/frequency trade off by
  using another transform e.g. wavelet, Wigner

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E.g. Multiresolution analysis (MRA)

• Analyses the signal at different frequencies with
  different resolutions
• By choosing
• Good time resolution and poor frequency
  resolution at high frequencies
• Good frequency resolution and poor time
  resolution at low frequencies
• This leads onto wavelet analysis

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Wavelet Transform
is the mother wavelet, which serves as the
prototype for successive wavelets of different
width depending on spectral component required
is translation i.e. the starting time of a
section of signal
is scale higher scale (e.g. s1) means more of
the signal looked at hence lower frequencies
seen, lower scale (e.g. s0.1) means less of the
signal looked at but in more detail.
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Summary

• A phase vocoder can shift or scale a musical
  spectrum, compress or expand signal duration and
  interpolate timbres all independently of other
  (unwanted) effects
• The phase vocoder implements the STFT (and its
  inverse), developed from the DFT
• Understanding of the SFTF is essential for proper
  control of a phase vocoder
• The concept of the phase vocoder can be extended
  by window closing, tracking or novel base
  transforms (e.g. wavelet).